definition of dualizing complex (from Stacks project)

Question

this question refers to definition of dualizing complex $\omega_A ^{\bullet}$ presented in Stacks as an object satisfying following properties:

(1)$ω^∙_A$ has finite injective dimension,

(2)$H^i(ω^∙_A)$ is a finite A-module for all i, and

(3) $A→RHomA(ω^∙_A,ω^∙_A)$ is a quasi-isomorphism.

my rudimetary question is what is the double complex $RHomA(ω^∙_A,ω^∙_A)$ and how is can it be calulated at each degree? what is the $l$-degree value of $RHomA(ω^∙_A,ω^∙_A)$ concretely? if we fix $A$-modules $M,N$ then $RHom(M,N)$ can be calculated in two way: take injective resolution $N \to I^{\bullet}$ for $N$(respectively projectlive resolution for $M$: $R^{\bullet} \to M$), apply the $Hom(M,-)$ functor (resp $Hom(,N)$) and take first homology. what is going on $RHomA(ω^∙_A,ω^∙_A)$?

Answer 1

In the Derived Hom section from the Stacks project, we have the following definition for the derived hom:

For $R$ a ring (denote by $D(R)$ the derived category of the category of $R$-modules) the derived hom is the functor $$ D(R)^\mathrm{op} \times D(R) \to D(R), \quad (M, L) \mapsto R\mathrm{Hom}_R(M, L) $$ and it is an internal hom in the sense that it is characterised by $$ \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K \otimes _ R^\mathbf {L} L, M) $$ for objects $K, L, M \in D(R)$. The derived hom itself is defined by $$ R\mathrm{Hom}(L, M) = \mathrm{Hom}^\bullet(I^\bullet, M^\bullet) $$ where $I^\bullet$ is a K-injective complex representing $M$ and $L^\bullet$ is a complex representing $L$. The hom complex here is defined by $$ \mathop{\mathrm{Hom}}\nolimits ^ n(I^\bullet , L^\bullet ) = \prod \nolimits _{n = p + q} \mathop{\mathrm{Hom}}\nolimits _ R(I^{-q}, L^ p). $$

In your case (I'm going to use $R$ instead of $A$ for the ring) $\omega_R^\bullet \in D(R)$ has finite injective dimension, so it's a finite complex and in particular is bounded below. Since the category of $R$-modules is an abelian category, by Lemma 13.29.4 from this Stacks article $\omega_R^\bullet$ is a K-injective complex. Taking $I^\bullet, L^\bullet = \omega_R^\bullet$ we can then calculate the derived hom at each degree.